Best Known (218−31, 218, s)-Nets in Base 2
(218−31, 218, 1094)-Net over F2 — Constructive and digital
Digital (187, 218, 1094)-net over F2, using
- 21 times duplication [i] based on digital (186, 217, 1094)-net over F2, using
- net defined by OOA [i] based on linear OOA(2217, 1094, F2, 31, 31) (dual of [(1094, 31), 33697, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2217, 16411, F2, 31) (dual of [16411, 16194, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2217, 16416, F2, 31) (dual of [16416, 16199, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2183, 16384, F2, 27) (dual of [16384, 16201, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2217, 16416, F2, 31) (dual of [16416, 16199, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(2217, 16411, F2, 31) (dual of [16411, 16194, 32]-code), using
- net defined by OOA [i] based on linear OOA(2217, 1094, F2, 31, 31) (dual of [(1094, 31), 33697, 32]-NRT-code), using
(218−31, 218, 3283)-Net over F2 — Digital
Digital (187, 218, 3283)-net over F2, using
- 21 times duplication [i] based on digital (186, 217, 3283)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2217, 3283, F2, 5, 31) (dual of [(3283, 5), 16198, 32]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2217, 16415, F2, 31) (dual of [16415, 16198, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(2217, 16416, F2, 31) (dual of [16416, 16199, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2183, 16384, F2, 27) (dual of [16384, 16201, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2217, 16416, F2, 31) (dual of [16416, 16199, 32]-code), using
- OOA 5-folding [i] based on linear OA(2217, 16415, F2, 31) (dual of [16415, 16198, 32]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2217, 3283, F2, 5, 31) (dual of [(3283, 5), 16198, 32]-NRT-code), using
(218−31, 218, 145412)-Net in Base 2 — Upper bound on s
There is no (187, 218, 145413)-net in base 2, because
- 1 times m-reduction [i] would yield (187, 217, 145413)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 210636 849386 595176 252060 536525 393042 871427 886274 643074 114703 126528 > 2217 [i]